Kirchhoff's laws

Kirchhoff’s laws give us relations that we base nodal and loop analysis around. They’re the closest thing to a universal law that we could have in electrical engineering.

The current law (KCL) tells us that the algebraic sum of the currents entering any node is zero.

$$\sum _{i=1}^n{i}_i=0$$

The voltage law (KVL) tells us that the algebraic sum of the voltages around any loop is zero.

$$\sum _{i=1}^n{v}_i=0$$

Both rules hold in the time, phasor, and frequency domains.

Proof

For a source-free magnetic field $\vec{H}$, we can express Gauss’ law as:

$$\nabla \times \vec{H}=\vec{J}(x,y,z)$$ $$\nabla \cdot \vec{J}=0$$

Applying the divergence theorem, we see that:

$$\oint {\oint }_S\vec{J}\cdot d\vec{S}={\iiint }_V\nabla \cdot \vec{J}dV=0$$

As in, the sum of the current fluxes entering or exiting a closed surface is zero, i.e., Kirchhoff’s current law for DC circuits.

For Faraday’s law with time-invariant electric fields (irrotational), we can express:

$$\nabla \times \vec{E}=0$$

Using Stokes’ theorem:

$${\oint }_C\vec{E}\cdot d\vec{l}={\iint }_A\nabla \times \vec{E}\cdot d\vec{S}=0$$

The circulation is zero, so the sum of potentials ($\vec{E}\cdot d\vec{l}$) around a closed loop (i.e., a closed circuit) is 0; this is the statement for Kirchhoff’s voltage law in DC. The property that the closed contour is equal to 0 is true for fields in electrostatic conditions.